Exponential functions and equations

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1. Exponential growth and decay: multiplying by a constant factor

Discover

Two ponds each start with 5 m² of algae. In pond A, the area grows by 5 m² each week. In pond B, the area doubles each week.

After 1 week both ponds have 10 m². Which pond has more algae after 4 weeks? Guess first.

  1. Pond A
  2. Pond B
  3. They have the same

Then check with a small table for weeks 0 to 4.

Key idea

When a quantity is multiplied by the same factor bb at each step, it changes exponentially.

Start aa, factor bb each step: after nn steps the amount is a⋅bna \cdot b^n

  • If b>1b > 1, the quantity grows (exponential growth).
  • If 0<b<10 < b < 1, it shrinks (exponential decay).
  • Losing p%p\% each step keeps (100−p)%(100 - p)\%, so the factor is 1−p1001 - \frac{p}{100}. Gaining p%p\% gives the factor 1+p1001 + \frac{p}{100}.

Worked example

A photo is 1000 pixels wide. Each time it is resized to 80% of its width. How wide is it after 3 resizes?

W=1000⋅0.8⋅0.8⋅0.8W = 1000 \cdot 0.8 \cdot 0.8 \cdot 0.8
Keeping 80% means multiplying by 0.8 each time.
W=1000⋅0.83W = 1000 \cdot 0.8^3
Three equal factors make a power.
W=512W = 512 pixels
0.8³ = 0.512.

Your turn, guided

Your turn: a plant is 3 cm tall, and its height triples every week. How tall is it after 4 weeks? Fill in the missing line.

h=3⋅34h = 3 \cdot 3^4
Start at 3 and multiply by 3 four times.
h=243h = 243 cm
3⁴ = 81, and 3 · 81 = 243.

Practice ★

  1. The amount of algae starts at 5 m² and is multiplied by 4 each week. Complete the table.
  2. The amount of algae starts at 2 m² and is multiplied by 2 each week. Complete the table.
  1. A culture starts with 7 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 3 weeks?
  2. A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 5 weeks?
  3. A culture starts with 3 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 5 weeks?
  4. A culture starts with 6 grams of algae. Each week, the amount is multiplied by 5. How many grams are there after 5 weeks?
  5. A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 3 weeks?

Watch out!

Practice ★★

  1. A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width W(t)W(t) after tt resizes.
  2. A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.
  3. A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width W(t)W(t) after tt resizes.
  4. A photo is 8000 pixels wide. Each time it is resized, its width loses 10%. Write the width W(t)W(t) after tt resizes.
  5. A photo is 4000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.

Exit ticket

Three questions to check you have got it.

  1. A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width W(t)W(t) after tt resizes.
  2. A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width W(t)W(t) after tt resizes.
  3. A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width W(t)W(t) after tt resizes.

Summary card

Exponential change: the same factor at every step

  • Start aa, factor bb per step: after nn steps, a⋅bna \cdot b^n.
  • Gain p%p\%: factor 1+p1001 + \frac{p}{100}. Lose p%p\%: factor 1−p1001 - \frac{p}{100}.

Watch out: adding the same amount each step is linear, not exponential.

2. Interpret exponential functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

f(x)=40⋅(12)xf(x) = 40 \cdot \left(\tfrac{1}{2}\right)^x. What is f(0)f(0)?

  1. 4040
  2. 00

Key idea

Read aa and bb from f(x)=a⋅bxf(x) = a \cdot b^x.

a=f(0)a = f(0) is the starting value (the yy-intercept). bb is the growth factor: b>1b > 1 growth, 0<b<10 < b < 1 decay. Negative inputs go back in time: f(−1)=a÷bf(-1) = a \div b.

Worked example

Graph through (0,3)(0, 3) and (1,12)(1, 12).

a=3a = 3
Starting value.
b=4b = 4
12 ÷ 3.
f(−1)=3÷4=0.75f(-1) = 3 \div 4 = 0.75
One step back.

Your turn, guided

Your turn: f(x)=50⋅5xf(x) = 50 \cdot 5^x. Find f(−1)f(-1).

1010
50 ÷ 5.

Practice ★

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  1. Is f(x)=4⋅(14)xf(x) = 4 \cdot \left(\tfrac{1}{ 4 }\right)^x exponential growth or decay?
    1. Decay: each step multiplies by 14\tfrac{1}{ 4 }
    2. Growth, because 4 is greater than 1
    3. A linear decrease

Watch out!

Practice ★★

  1. The graph of an exponential function passes through (0,8)(0, 8) and (1,16)(1, 16). Write f(x)f(x).
  2. The graph of an exponential function passes through (0,7)(0, 7) and (1,21)(1, 21). Write f(x)f(x).

Exit ticket

Three questions to check you have got it.

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  3. f(x)=27⋅3xf(x) = 27 \cdot 3^x. Find f(−1)f(-1).

Summary card

Reading exponential functions

aa: value at x=0x = 0; bb: factor per step; b−1=1/bb^{-1} = 1/b.

Watch out: negative exponent, positive value.

3. Rational exponents in exponential functions

Warm-up

Three quick questions from earlier lessons.

Discover

A bacteria population doubles every day. By what factor does it grow in half a day?

Hint: whatever the factor ff is, growing for two half-days must double the population, so f⋅f=2f \cdot f = 2.

Key idea

Exponents that are fractions are roots.

b1n=bnb^{\frac{1}{n}} = \sqrt[n]{b} and bmn=(bn)mb^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m

So the half-day factor is 212=2≈1.412^{\frac12} = \sqrt{2} \approx 1.41. A negative exponent means a reciprocal: b−r=1brb^{-r} = \dfrac{1}{b^{r}}.

Worked example

Evaluate 8238^{\frac{2}{3}}.

823=(83)28^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2
The denominator 3 means a cube root; the numerator 2 means a square.
=22= 2^2
The cube root of 8 is 2.
=4= 4
Square it.

Your turn, guided

Your turn: evaluate 272327^{\frac{2}{3}}. Fill in the missing line.

2723=(273)2=3227^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2
Cube root, then square.
=9= 9
3² = 9.

Practice ★

  1. Evaluate 365236^{ \frac{ 5 }{2} }.
  2. Evaluate 813281^{ \frac{ 3 }{2} }.
  3. Evaluate 815281^{ \frac{ 5 }{2} }.
  4. Evaluate 4324^{ \frac{ 3 }{2} }.
  5. Evaluate 495249^{ \frac{ 5 }{2} }.

Watch out!

Practice ★★

  1. Evaluate 125−23125^{ -\frac{ 2 }{3} }.
  2. Evaluate 125−13125^{ -\frac{ 1 }{3} }.
  3. Evaluate 64−1364^{ -\frac{ 1 }{3} }.
  4. Evaluate 8−138^{ -\frac{ 1 }{3} }.
  5. Evaluate 27−1327^{ -\frac{ 1 }{3} }.

Exit ticket

Three questions to check you have got it.

  1. Evaluate 125−23125^{ -\frac{ 2 }{3} }.
  2. Evaluate 125−13125^{ -\frac{ 1 }{3} }.
  3. Evaluate 64−1364^{ -\frac{ 1 }{3} }.

Summary card

Rational exponents

  • b1n=bnb^{\frac{1}{n}} = \sqrt[n]{b}; bmn=(bn)mb^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m.
  • b−r=1brb^{-r} = \dfrac{1}{b^{r}}: the value stays positive.
  • A factor bb per unit means a factor b12b^{\frac12} per half-unit.

4. Write an exponential function from a description or two points

Warm-up

Three quick questions from earlier lessons.

Discover

A table shows a function: f(0)=4f(0) = 4, f(1)=12f(1) = 12, f(2)=36f(2) = 36, f(3)=108f(3) = 108.

What do you multiply by to go from one value to the next? Can you write one formula that gives all four values?

Key idea

An exponential function can be written

f(x)=a⋅bxf(x) = a \cdot b^x, where a=f(0)a = f(0) and bb is the factor for each increase of 1 in xx

If the quantity is multiplied by bb every nn units instead, divide the input by nn: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Worked example

Write ff for the table above.

a=4a = 4
The starting value is f(0).
b=3b = 3
Divide consecutive values: 12 ÷ 4 = 3.
f(x)=4⋅3xf(x) = 4 \cdot 3^x
Put them in the form a · b^x.

Your turn, guided

Your turn: f(0)=5f(0) = 5 and f(1)=10f(1) = 10. Fill in the missing line.

a=5a = 5, b=2b = 2
10 ÷ 5 = 2.
f(x)=5⋅2xf(x) = 5 \cdot 2^x
Use the form a · b^x.

Practice ★

  1. Plot the points of the graph of f(x)=4⋅2xf(x) = 4 \cdot 2^x at x=0x = 0 and x=1x = 1.
  2. Plot the points of the graph of f(x)=3⋅4xf(x) = 3 \cdot 4^x at x=0x = 0 and x=1x = 1.
  1. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 5. Write f(x)f(x).
  2. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  3. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).
  4. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  5. An exponential function ff has f(0)=4f(0) = 4, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).

Watch out!

Practice ★★

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  4. A population is 8 thousand at t=0t = 0 and is multiplied by 2 every 2 years. Write the population P(t)P(t), in thousands, after tt years.
  5. A population is 9 thousand at t=0t = 0 and is multiplied by 3 every 5 years. Write the population P(t)P(t), in thousands, after tt years.

Exit ticket

Three questions to check you have got it.

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.

Summary card

Writing an exponential function

  • f(x)=a⋅bxf(x) = a \cdot b^x: a=f(0)a = f(0), b=f(1)f(0)b = \dfrac{f(1)}{f(0)}.
  • Multiplied by bb every nn units: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Watch out: the starting value is f(0)f(0), and it goes in front, not in the base.

5. What is a logarithm? Logarithms as exponents

Warm-up

Three quick questions from earlier lessons.

Discover

Here is part of a table of powers of 10: 100=110^0 = 1, 101=1010^1 = 10, 102=10010^2 = 100, 103=100010^3 = 1000.

Solve 10x=100010^x = 1000. Now try 10x=5010^x = 50: between which two whole numbers is xx?

Key idea

A logarithm is an exponent. log⁡bx\log_b x answers the question: "bb to what power gives xx?"

log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x (with b>0b > 0, b≠1b \ne 1, x>0x > 0)

Written without a base, log⁡x\log x means log⁡10x\log_{10} x. So the solution of 10x=5010^x = 50 is x=log⁡50≈1.7x = \log 50 \approx 1.7.

Worked example

Evaluate log⁡232\log_2 32.

2y=322^{y} = 32
Ask: 2 to what power gives 32?
y=5y = 5
2⁵ = 32.
log⁡232=5\log_2 32 = 5
So the logarithm is 5.

Your turn, guided

Your turn: evaluate log⁡381\log_3 81. Fill in the missing line.

3y=813^{y} = 81
Ask: 3 to what power gives 81?
log⁡381=4\log_3 81 = 4
3⁴ = 81.

Practice ★

  1. Evaluate log⁡416\log_{ 4 } 16.
  2. Evaluate log⁡525\log_{ 5 } 25.
  3. Evaluate log⁡4256\log_{ 4 } 256.
  4. Evaluate log⁡464\log_{ 4 } 64.
  5. Evaluate log⁡28\log_{ 2 } 8.

Watch out!

Practice ★★

  1. Match each logarithm statement with the same statement written as a power.
    1. log⁡8y=2\log_{ 8 } y = 2
    2. log⁡8y=3\log_{ 8 } y = 3
    3. log⁡8y=−1\log_{ 8 } y = -1
    4. log⁡8y=12\log_{ 8 } y = \tfrac{1}{2}
    1. y=82y = 8^{2}
    2. y=83y = 8^{3}
    3. y=18y = \dfrac{1}{ 8 }
    4. y=8y = \sqrt{ 8 }
  2. Match each logarithm statement with the same statement written as a power.
    1. log⁡3y=2\log_{ 3 } y = 2
    2. log⁡3y=3\log_{ 3 } y = 3
    3. log⁡3y=−1\log_{ 3 } y = -1
    4. log⁡3y=12\log_{ 3 } y = \tfrac{1}{2}
    1. y=32y = 3^{2}
    2. y=33y = 3^{3}
    3. y=13y = \dfrac{1}{ 3 }
    4. y=3y = \sqrt{ 3 }
  1. Solve 6x=216^x = 21. Write the exact answer using a logarithm.
  2. Solve 7x=167^x = 16. Write the exact answer using a logarithm.
  3. Solve 2x=312^x = 31. Write the exact answer using a logarithm.
  4. Solve 7x=157^x = 15. Write the exact answer using a logarithm.
  5. Solve 2x=372^x = 37. Write the exact answer using a logarithm.

Exit ticket

Three questions to check you have got it.

  1. Solve 6x=216^x = 21. Write the exact answer using a logarithm.
  2. Solve 7x=167^x = 16. Write the exact answer using a logarithm.
  3. Solve 2x=312^x = 31. Write the exact answer using a logarithm.

Summary card

Logarithms are exponents

  • log⁡bx=y\log_b x = y means by=xb^y = x.
  • log⁡x\log x means log⁡10x\log_{10} x.
  • The solution of bx=cb^x = c is x=log⁡bcx = \log_b c.

Watch out: log⁡bx\log_b x is never x÷bx \div b.

6. The number e and exponential functions with base e

Warm-up

Three quick questions from earlier lessons.

Discover

Compute (1+1x)x\left(1 + \frac{1}{x}\right)^x for x=1x = 1, 1010, 100100 and 10001000 with a calculator.

What do you notice as xx gets larger?

Key idea

The values approach a number called e≈2.718e \approx 2.718. Functions with base ee describe continuous growth.

Continuous growth at rate rr (as a decimal): A(t)=A0 ertA(t) = A_0 \, e^{rt}

In f(t)=a ektf(t) = a \, e^{kt}, increasing tt by 1 multiplies ff by eke^{k}.

Worked example

The area of a mold is a(t)=10eta(t) = 10e^{t} mm² after tt days. By what factor does it grow each day, and what is the area after 2 days?

Daily factor: e≈2.718e \approx 2.718
Here k = 1, so each day multiplies the area by e¹.
a(2)=10e2a(2) = 10e^{2}
Substitute t = 2.
a(2)≈73.9a(2) \approx 73.9 mm²
e² ≈ 7.389.

Your turn, guided

Your turn: 500 dollars grows continuously at 4% per year. Fill in the missing line.

r=0.04r = 0.04
4% as a decimal.
A(t)=500e0.04tA(t) = 500e^{0.04t}
Use A₀ e^(rt).

Practice ★

  1. An account holds 400 dollars and grows continuously at 3% per year. Write the amount A(t)A(t) after tt years.
  2. An account holds 100 dollars and grows continuously at 8% per year. Write the amount A(t)A(t) after tt years.
  3. An account holds 300 dollars and grows continuously at 3% per year. Write the amount A(t)A(t) after tt years.
  4. An account holds 600 dollars and grows continuously at 6% per year. Write the amount A(t)A(t) after tt years.
  5. An account holds 500 dollars and grows continuously at 5% per year. Write the amount A(t)A(t) after tt years.

Watch out!

Practice ★★

  1. The area of mold is f(t)=2e4tf(t) = 2e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  2. The area of mold is f(t)=3e1tf(t) = 3e^{ 1t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  3. The area of mold is f(t)=7e4tf(t) = 7e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  4. The area of mold is f(t)=2e2tf(t) = 2e^{ 2t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  5. The area of mold is f(t)=3e4tf(t) = 3e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.

Exit ticket

Three questions to check you have got it.

  1. The area of mold is f(t)=2e4tf(t) = 2e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  2. The area of mold is f(t)=3e1tf(t) = 3e^{ 1t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  3. The area of mold is f(t)=7e4tf(t) = 7e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.

Summary card

The number ee

  • e≈2.718e \approx 2.718, the limit of (1+1x)x\left(1 + \frac{1}{x}\right)^x.
  • Continuous growth at rate rr: A(t)=A0 ertA(t) = A_0 \, e^{rt}.
  • In a ekta \, e^{kt}, the factor per unit of tt is eke^{k}.

Watch out: "continuously at rr" is not "by 1+r1 + r once a year".

7. Solve equations of the form e^(kx) = a

Warm-up

Three quick questions from earlier lessons.

Discover

A colony of bacteria doubles every hour, so after tt hours there are e0.69te^{0.69t} times as many. When will there be 7 times as many? Try to estimate before reading on: between which two whole hours?

Key idea

To undo exe^x, use its inverse function: ln⁡\ln.

ex=a  ⟺  x=ln⁡ae^x = a \iff x = \ln a, for a>0a > 0

Since ln⁡(ex)=x\ln(e^{x}) = x, taking ln⁡\ln of both sides "removes" the exponential.

Worked example

Solve e2x=7e^{2x} = 7.

2x=ln⁡72x = \ln 7
Apply ln to both sides: ln undoes e.
x=ln⁡72x = \dfrac{\ln 7}{2}
Divide both sides by 2.

Your turn, guided

Your turn: solve e3x=5e^{3x} = 5. Fill in the missing line.

3x=ln⁡53x = \ln 5
Apply ln to both sides.
x=ln⁡53x = \dfrac{\ln 5}{3}
Divide both sides by 3.

Practice ★

  1. Solve e5x=25e^{ 5x } = 25.
  2. Solve e5x=7e^{ 5x } = 7.
  3. Solve e5x=9e^{ 5x } = 9.
  4. Solve e3x=15e^{ 3x } = 15.
  5. Solve e5x=10e^{ 5x } = 10.

Watch out!

Practice ★★

  1. Solve e4x+4=12e^{ 4x + 4 } = 12.
  2. Solve e6x+5=37e^{ 6x + 5 } = 37.
  3. Solve e2x+7=16e^{ 2x + 7 } = 16.
  4. Solve e3x+6=27e^{ 3x + 6 } = 27.
  5. Solve e6x+7=39e^{ 6x + 7 } = 39.

Exit ticket

Three questions to check you have got it.

  1. Solve e4x+4=12e^{ 4x + 4 } = 12.
  2. Solve e6x+5=37e^{ 6x + 5 } = 37.
  3. Solve e2x+7=16e^{ 2x + 7 } = 16.

Summary card

Solving ekx=ae^{kx} = a (with a>0a > 0)

  1. Take ln⁡\ln of both sides: kx=ln⁡akx = \ln a.
  2. Divide by kk: x=ln⁡akx = \dfrac{\ln a}{k}.

Watch out: ee is not a factor. Never divide by ee.

One more step is optional: a challenge. Try it, or finish the lesson now.

Challenge ★★★

  1. Solve 3e4x+6=60 3e^{ 4x } + 6 = 60.
  2. Solve 5e2x+6=25 5e^{ 2x } + 6 = 25.
  3. Solve 4e2x+8=36 4e^{ 2x } + 8 = 36.

8. Logarithmic functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

The graph of y=2xy = 2^x passes through (0,1)(0, 1), (1,2)(1, 2) and (3,8)(3, 8). Swap the coordinates of each point.

Which function has a graph through the swapped points?

Key idea

y=log⁡bxy = \log_b x is the inverse of y=bxy = b^x: its graph is the reflection of y=bxy = b^x in the line y=xy = x.

The graph of y=log⁡bxy = \log_b x passes through (1,0)(1, 0) and (b,1)(b, 1), has domain x>0x > 0 and a vertical asymptote x=0x = 0

For y=log⁡b(x−h)y = \log_b(x - h), everything shifts hh units right: the asymptote is x=hx = h and the graph crosses the xx-axis where x−h=1x - h = 1.

Worked example

Where does y=log⁡2(x−3)y = \log_2(x - 3) cross the xx-axis?

log⁡2(x−3)=0\log_2(x - 3) = 0
On the x-axis, y = 0.
x−3=1x - 3 = 1
A logarithm is 0 when its input is 1.
x=4x = 4
Add 3.

Your turn, guided

Your turn: where does y=ln⁡(x−5)y = \ln(x - 5) cross the xx-axis? Fill in the missing line.

x−5=1x - 5 = 1
ln of the input is 0 when the input is 1.
x=6x = 6
Add 5.

Practice ★

  1. Which statements are true about the graph of y=log⁡9xy = \log_{ 9 } x? Tick all that are true.
    1. It passes through (1, 0).
    2. It passes through (9, 1).
    3. It passes through (0, 1).
    4. It has the horizontal asymptote y = 0.
    5. It is defined only for x > 0.
  2. Which statements are true about the graph of y=log⁡2xy = \log_{ 2 } x? Tick all that are true.
    1. It passes through (1, 0).
    2. It passes through (2, 1).
    3. It passes through (0, 1).
    4. It has the horizontal asymptote y = 0.
    5. It is defined only for x > 0.
  1. The graph of y=log⁡bxy = \log_b x passes through the point (9,2)(9, 2). What is bb?
  2. The graph of y=log⁡bxy = \log_b x passes through the point (81,4)(81, 4). What is bb?
  3. The graph of y=log⁡bxy = \log_b x passes through the point (8,3)(8, 3). What is bb?
  4. The graph of y=log⁡bxy = \log_b x passes through the point (625,4)(625, 4). What is bb?
  5. The graph of y=log⁡bxy = \log_b x passes through the point (64,3)(64, 3). What is bb?

Watch out!

Practice ★★

  1. Here is the graph of y=log⁡3(x−5)y = \log_{ 3 }(x - 5). Tap the point where it crosses the xx-axis.
  2. Here is the graph of y=log⁡2(x−4)y = \log_{ 2 }(x - 4). Tap the point where it crosses the xx-axis.
  1. What is the domain of y=ln⁡(x−6)y = \ln(x - 6)? Give it as an interval.
  2. What is the domain of y=ln⁡(x−9)y = \ln(x - 9)? Give it as an interval.
  3. What is the domain of y=ln⁡(x−7)y = \ln(x - 7)? Give it as an interval.
  4. What is the domain of y=ln⁡(x−2)y = \ln(x - 2)? Give it as an interval.
  5. What is the domain of y=ln⁡(x−8)y = \ln(x - 8)? Give it as an interval.

Exit ticket

Three questions to check you have got it.

  1. Where does the graph of y=ln⁡(x−3)y = \ln(x - 3) cross the xx-axis? Give the value of xx.
  2. Where does the graph of y=ln⁡(x−6)y = \ln(x - 6) cross the xx-axis? Give the value of xx.
  3. Where does the graph of y=ln⁡(x−7)y = \ln(x - 7) cross the xx-axis? Give the value of xx.

Summary card

Logarithmic functions

  • y=log⁡bxy = \log_b x is the inverse of y=bxy = b^x.
  • Through (1,0)(1, 0) and (b,1)(b, 1); domain x>0x > 0; asymptote x=0x = 0.
  • y=log⁡b(x−h)y = \log_b(x - h): asymptote x=hx = h; crosses the xx-axis at x=h+1x = h + 1.

Answers: Exponential functions and equations

1. Exponential growth and decay: multiplying by a constant factor

A

  1. 20,80,32020, 80, 320
  2. 4,8,164, 8, 16

B

  1. 189189
  2. 160160
  3. 729729
  4. 1875018750
  5. 4040

C

  1. 8000(35)t8000 \left(\frac{3}{5}\right)^{t}
  2. 2000(710)t2000 \left(\frac{7}{10}\right)^{t}
  3. 2000(34)t2000 \left(\frac{3}{4}\right)^{t}
  4. 8000(910)t8000 \left(\frac{9}{10}\right)^{t}
  5. 4000(710)t4000 \left(\frac{7}{10}\right)^{t}

D

  1. 8000(35)t8000 \left(\frac{3}{5}\right)^{t}
  2. 2000(710)t2000 \left(\frac{7}{10}\right)^{t}
  3. 2000(34)t2000 \left(\frac{3}{4}\right)^{t}

2. Interpret exponential functions and their graphs

A

  1. 66
  2. 66

B

  1. Decay: each step multiplies by 14\tfrac{1}{ 4 }

C

  1. 8⋅2x8 \cdot 2^{x}
  2. 7⋅3x7 \cdot 3^{x}

D

  1. 66
  2. 66
  3. 99

3. Rational exponents in exponential functions

A

  1. 77767776
  2. 729729
  3. 5904959049
  4. 88
  5. 1680716807

B

  1. 125\frac{1}{25}
  2. 15\frac{1}{5}
  3. 14\frac{1}{4}
  4. 12\frac{1}{2}
  5. 13\frac{1}{3}

C

  1. 125\frac{1}{25}
  2. 15\frac{1}{5}
  3. 14\frac{1}{4}

4. Write an exponential function from a description or two points

A

  1. (0,4), (1,8)(0, 4),\ (1, 8)
  2. (0,3), (1,12)(0, 3),\ (1, 12)

B

  1. 9⋅5x9 \cdot 5^{x}
  2. 9⋅4x9 \cdot 4^{x}
  3. 6⋅2x6 \cdot 2^{x}
  4. 6⋅4x6 \cdot 4^{x}
  5. 4⋅2x4 \cdot 2^{x}

C

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}
  4. 8⋅2t28 \cdot 2^{\frac{t}{2}}
  5. 9⋅3t59 \cdot 3^{\frac{t}{5}}

D

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}

5. What is a logarithm? Logarithms as exponents

A

  1. 22
  2. 22
  3. 44
  4. 33
  5. 33

B

  1. log⁡8y=2\log_{ 8 } y = 2 → y=82y = 8^{2} · log⁡8y=3\log_{ 8 } y = 3 → y=83y = 8^{3} · log⁡8y=−1\log_{ 8 } y = -1 → y=18y = \dfrac{1}{ 8 } · log⁡8y=12\log_{ 8 } y = \tfrac{1}{2} → y=8y = \sqrt{ 8 }
  2. log⁡3y=2\log_{ 3 } y = 2 → y=32y = 3^{2} · log⁡3y=3\log_{ 3 } y = 3 → y=33y = 3^{3} · log⁡3y=−1\log_{ 3 } y = -1 → y=13y = \dfrac{1}{ 3 } · log⁡3y=12\log_{ 3 } y = \tfrac{1}{2} → y=3y = \sqrt{ 3 }

C

  1. ln⁡(21)ln⁡(6)\frac{\ln{\left(21 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(16)ln⁡(7)\frac{\ln{\left(16 \right)}}{\ln{\left(7 \right)}}
  3. ln⁡(31)ln⁡(2)\frac{\ln{\left(31 \right)}}{\ln{\left(2 \right)}}
  4. ln⁡(15)ln⁡(7)\frac{\ln{\left(15 \right)}}{\ln{\left(7 \right)}}
  5. ln⁡(37)ln⁡(2)\frac{\ln{\left(37 \right)}}{\ln{\left(2 \right)}}

D

  1. ln⁡(21)ln⁡(6)\frac{\ln{\left(21 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(16)ln⁡(7)\frac{\ln{\left(16 \right)}}{\ln{\left(7 \right)}}
  3. ln⁡(31)ln⁡(2)\frac{\ln{\left(31 \right)}}{\ln{\left(2 \right)}}

6. The number e and exponential functions with base e

A

  1. 400e3t100400 e^{\frac{3 t}{100}}
  2. 100e2t25100 e^{\frac{2 t}{25}}
  3. 300e3t100300 e^{\frac{3 t}{100}}
  4. 600e3t50600 e^{\frac{3 t}{50}}
  5. 500et20500 e^{\frac{t}{20}}

B

  1. e4e^{4}
  2. ee
  3. e4e^{4}
  4. e2e^{2}
  5. e4e^{4}

C

  1. e4e^{4}
  2. ee
  3. e4e^{4}

7. Solve equations of the form e^(kx) = a

A

  1. ln⁡(25)5\frac{\ln{\left(25 \right)}}{5}
  2. ln⁡(7)5\frac{\ln{\left(7 \right)}}{5}
  3. ln⁡(9)5\frac{\ln{\left(9 \right)}}{5}
  4. ln⁡(15)3\frac{\ln{\left(15 \right)}}{3}
  5. ln⁡(10)5\frac{\ln{\left(10 \right)}}{5}

B

  1. −1+ln⁡(12)4-1 + \frac{\ln{\left(12 \right)}}{4}
  2. −56+ln⁡(37)6- \frac{5}{6} + \frac{\ln{\left(37 \right)}}{6}
  3. −72+ln⁡(16)2- \frac{7}{2} + \frac{\ln{\left(16 \right)}}{2}
  4. −2+ln⁡(27)3-2 + \frac{\ln{\left(27 \right)}}{3}
  5. −76+ln⁡(39)6- \frac{7}{6} + \frac{\ln{\left(39 \right)}}{6}

C

  1. −1+ln⁡(12)4-1 + \frac{\ln{\left(12 \right)}}{4}
  2. −56+ln⁡(37)6- \frac{5}{6} + \frac{\ln{\left(37 \right)}}{6}
  3. −72+ln⁡(16)2- \frac{7}{2} + \frac{\ln{\left(16 \right)}}{2}

D

  1. ln⁡(18)4\frac{\ln{\left(18 \right)}}{4}
  2. ln⁡(195)2\frac{\ln{\left(\frac{19}{5} \right)}}{2}
  3. ln⁡(7)2\frac{\ln{\left(7 \right)}}{2}

8. Logarithmic functions and their graphs

A

  1. It passes through (1, 0). · It passes through (9, 1). · It is defined only for x > 0.
  2. It passes through (1, 0). · It passes through (2, 1). · It is defined only for x > 0.

B

  1. 33
  2. 33
  3. 22
  4. 55
  5. 44

C

  1. (6,0)(6, 0)
  2. (5,0)(5, 0)

D

  1. (6,∞)(6, \infty)
  2. (9,∞)(9, \infty)
  3. (7,∞)(7, \infty)
  4. (2,∞)(2, \infty)
  5. (8,∞)(8, \infty)

E

  1. 44
  2. 77
  3. 88